What are your favourite classical physics puzzles?

[etotheipi]

^ The title is pretty self-explanatory, wondered if people would like to share some fun problems that you can snuggle up to on a cold Winter's evening? Any difficulty level, but bonus points for problems which yield to elegant and/or creative solutions!

Here's an example:

Consider three co-planar point masses $m_1, \, m_2$ and $m_3$, interacting only under their mutual gravitation. Let the distances between the masses be $r_{12}, \, r_{13}$ and $r_{23}$. Consider an axis perpendicular to the plane containing the masses, passing through the centre of mass of the system. If the system rotates about this axis as a rigid body, then what must be the angular speed $\omega$ of this rotation, and the relationship between $r_{12}, \, r_{13}$ and $r_{23}$?

 

[phinds]

Your spoiler link goes to the black hole

 

[berkeman]

Your spoiler link goes to the black hole

Looks to be fixed now...

 

[berkeman]

Looks to be fixed now...

https://www.physicsforums.com/attachments/277234

Although, come to think of it, I've never seen $e^{i\pi}$ post hand-written equations. I'm thinking somebody has hacked his account...

 

[etotheipi]

Although, come to think of it, I've never seen $e^{i\pi}$ post hand-written equations. I'm thinking somebody has hacked his account...

Cmon man it's 1:47 AM and you're expecting me to type up all that $\LaTeX$?

 

[A.T.]

The title is pretty self-explanatory, wondered if people would like to share some fun problems that you can snuggle up to on a cold Winter's evening?

What is the shape of an asteroid of given uniform density and given total mass such that the force of gravity is maximal for one point on its surface? Compare the maximal gravity with that of a sphere of the same mass and density

Solution:
http://kirkmcd.princeton.edu/examples/maximal_gravity.pdf

More:
https://www.tau.ac.il/~kantor/QUIZ/

 

[wrobel]

There is a curve $\gamma$ drawn on a plane $P$. The curve is given by its curvature $k=k(s)$, here $s$ is the arc-length parameter. A ball of radius $r$ is placed on the plane and one rolls the ball such that the contact point follows the curve $\gamma$ and the angular velocity of the ball is parallel to the plane. The ball rolls without slipping. The contact point draws a curve $\Gamma$ on the ball. Find the curvature $K(s)$ of this curve. Find the torsion $\tau(s)$ of $\Gamma$.

 

[cmb]

Any puzzle that starts off with 'Consider a perfectly spherical cow ...'. :)

 

[Demystifier]

Why does entropy increase? (Explanation sought within classical statistical mechanics.)

 

[vanhees71]

Why doesn't kinetic theory within classical statistics leading to Boltzmann's H theorem convince you?

 

[atyy]

I still don't understand (intuitively), why in rolling without slipping, the bottom of the wheel is stationary!

 

[atyy]

Why doesn't kinetic theory within classical statistics leading to Boltzmann's H theorem convince you?

Assumption of molecular chaos?

 

[vanhees71]

Sure, that's at the heart of the increasing entropy. You need coarse-graining, i.e., you must neglect some information to get an increasing entropy. I think that's a pretty intuitive necessity and that's why I find this a very convincing argument for why entropy should not decrease.

 

[atyy]

Sure, that's at the heart of the increasing entropy. You need coarse-graining, i.e., you must neglect some information to get an increasing entropy. I think that's a pretty intuitive necessity and that's why I find this a very convincing argument for why entropy should not decrease.

OK, but then if I use this intuition, I don't even need the H-theorem to believe it ...

 

[vanhees71]

Sure, if there is one really intuitive physical law it's the increase of entropy. I can observe it everyday on my desk ;-)).

 

[atyy]

http://archive.boston.com/realestate/galleries/springsweep/13.htm
Alan Guth's office!

Here's an article with another photo.

 

[Demystifier]

Why doesn't kinetic theory within classical statistics leading to Boltzmann's H theorem convince you?

The H theorem (through the molecular chaos assumption) secretly assumes what it wants to prove, so it only shifts the problem, rather than solves it. Sure, the assumption is intuitive because it's in accordance with our everyday experience, but so is the law of increase of entropy. The right formulation of the problem is - why was the entropy low far in the past?

 

[Dale]

secretly assumes what it wants to prove

Isn’t that true of every proof? In fact, isn’t the point of every proof to show exactly how the the secret assumptions are hidden within the stated assumptions?

The question isn’t whether or not the conclusion is secretly assumed, it always is. The question is only if the stated assumptions are acceptable.

 

[Demystifier]

Isn’t that true of every proof? In fact, isn’t the point of every proof to show exactly how the the secret assumptions are hidden within the stated assumptions?

The question isn’t whether or not the conclusion is secretly assumed, it always is. The question is only if the stated assumptions are acceptable.

In some sense you are absolutely right, and in some sense you are deeply wrong. But the sense in which you are wrong is so deep that I can't explain why exactly are you wrong. It would be an interesting separate thread.

 

[vanhees71]

I think the argument that you cut the BBGKY hierarchy at the one-particle distribution function level by assuming the two-particle distribution to factor in two one-particle distribution functions, i.e., neglecting correlations between particle pairs (and of course also all other higher $n$-particle correlations) using the molecular chaos ansatz and thus just throw away information which you cannot resolve very plausible. That this implies an increase of entropy (on average) is of course indeed obvious when you take entropy as the measure of missing information a la Shannon and Jaynes.

What I think is wrong is the argument that this establishes an arrow of time. This arrow of time is really explicitly used in the argument, because we use the "causal arrow of time" to define a distinction between past and future, i.e., all we get is that the "causal arrow of time" implies the "thermodynamic arrow of time" making the molecular-chaos assumption in cutting the BBGKY hierarchy.

This view becomes more clear when using many-body QFT to derive the Boltzmann equation. There you explicitly see, how through coarse-graining you neglect dynamical information on the two-particle correlation level. See, e.g.,

K. Chou, Z. Su, B. Hao and L. Yu, Equilibrium and
Nonequilibrium Formalisms made unified, Phys. Rept. 118, 1
(1985), https://doi.org/10.1016/0370-1573(85)90136-X.

 

[Demystifier]

What I think is wrong is the argument that this establishes an arrow of time. This arrow of time is really explicitly used in the argument, because we use the "causal arrow of time" to define a distinction between past and future, i.e., all we get is that the "causal arrow of time" implies the "thermodynamic arrow of time" making the molecular-chaos assumption in cutting the BBGKY hierarchy.

So does the thermodynamic arrow of time explain the causal arrow of time, or is it the other way around?

 

[Dale]

In some sense you are absolutely right, and in some sense you are deeply wrong.

Well, of course I intended it in the “absolutely right” sense and not in the “deeply wrong” sense

My point is more that you probably simply disagree with one or more of assumptions. Perhaps the molecular chaos assumption. And IMO, that is fair. One who wishes to convince another must start with assumptions that the other accepts.

 

[vanhees71]

It's the other way round: We assume (always!) a causal arrow of time. It's hidden in the words we use to describe a situation. It's in the solutions we pick out to describe physics from all other possible solutions, e.g., in classical mechanics we impose initial conditions for the positions and velocities or in electrodynamics for $\vec{E}$ and $\vec{B}$.

The latter example is also interesting: There we implicitly also use the causal arrow of time by using the retarded solutions for physical quantities (i.e., the fields rather than the potentials!), leading to the usual outgoing waves from the sources rather than the incoming ones which are possible solutions of Maxwell's equations (as it must be, because as the time-reversed solutions they fulfill the Maxwell equations as the outgoing solutions do, because of $T$-invariance of these equations). The incoming solution is simply hard to observe, because it's hard to impose the corresponding initial conditions: Some spherical-wave at a far distance being absorbed by the charges and currents moving in the right way. In this sense the "electromagnetic arrow of time" follows from the "causal arrow of time" in the same way as the "thermodynamical arrow of time", and in this sense the electromagnetic and thermodynamical arrows of time are of the same nature.

 

[Demystifier]

@vanhees71 And where does the causal arrow of time come from?

 

[vanhees71]

That's indeed one of the very fundamental assumptions we make. Of course, without causality there'd be no natural sciences to begin with, because there'd be no laws to study.

 

[Demystifier]

That's indeed one of the very fundamental assumptions we make. Of course, without causality there'd be no natural sciences to begin with, because there'd be no laws to study.

Is it a physical or philosophical issue?

 

[vanhees71]

I'd say it's just logics. If there are no laws to be discovered and consequently nothing leading to a science that describes them.

 

[Charles Link]

^ The title is pretty self-explanatory, wondered if people would like to share some fun problems that you can snuggle up to on a cold Winter's evening? Any difficulty level, but bonus points for problems which yield to elegant and/or creative solutions!

Here's an example:

See https://www.physicsforums.com/threads/physics-challenge-by-quantumquest-1.909391/#post-5729713

 

[Charles Link]

See https://www.physicsforums.com/threads/micromass-big-october-challenge.887447/
The first puzzle is one that I contributed. (It is my own invention). I had showed it to micromass and he liked it so much that he included it in his monthly challenge.
See also https://www.physicsforums.com/threa...ch-result-in-the-cornu-spiral-comment.902917/

 

[guv]

There is a curve $\gamma$ drawn on a plane $P$. The curve is given by its curvature $k=k(s)$, here $s$ is the arc-length parameter. A ball of radius $r$ is placed on the plane and one rolls the ball such that the contact point follows the curve $\gamma$ and the angular velocity of the ball is parallel to the plane. The ball rolls without slipping. The contact point draws a curve $\Gamma$ on the ball. Find the curvature $K(s)$ of this curve. Find the torsion $\tau(s)$ of $\Gamma$.

@wrobel I am not sure what you mean by the curve traced by contact point. Are you referring to 'a' contact point labelled on the ball. And you are looking at the curve traced out by 'that' contact point. Otherwise isn't the curve traced out by all the contact points at different instants in time exactly $\gamma$?

 

[guv]

@wrobel I missed the part $\omega$ is parallel to the plane.

 

[A.T.]

wondered if people would like to share some fun problems that you can snuggle up to on a cold Winter's evening?

There is a sailing race down a river which flows at 10 km/h.

- On the first day the wind relative to the ground blows at 10 km/h parallel to the river in the downstream direction.

- On the second day there is no wind relative to the ground.

During which day can the sailboats make faster progress down the river?

 

[Charles Link]

I believe the correct answer for @A.T. 's sailing puzzle is the second day.
(I've done a little sailing, and to head into the direction the wind is coming from, you zigzag toward the destination).